It looks like flint, possibly banded flint with the ammonite/trilobite- like markings. Lyme regis consists of lower jurassic rocks only, and trilobites didn't live at this time. Of corse, there's a chance the markings could be fossils, but I think Mike may know as he is a good Geologist.

Keep looking though; Lyme Regis has some super fossils, including marine reptiles and occasionally even dinosaurs.

Dan's right - it is banded flint. These objects are often mistaken for fossils.

We, as humans, have evolved to be good at recognizing patterns. It helped us survive and develop over tens of thousands of years - it is how we know, just by looking, what is- or what might be food, for instance. We recognize many shapes as being organic in origin. Unfortunately, some shapes in nature are superficially ambiguous - they could be organic or mineral in origin. In these situations, we rely on context and other knowledge to understand the object. Both those factors require experience. And if one is missing some experience, it is possible to make the wrong deduction as to the origin.

That is your problem here. That is: you noticed a pattern, and recognized it as being akin to part of a trilobite. Trouble is, you have never seen banded flint, so you didn't know that is also a possibility.

But now you do, so you will have that extra experience ready for next time.